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Math 2224 Multivariable Calc – Sec. 12.4: The Chain Rule
I.
Review from math 1205: The Chain Rule for functions of single variables
A. The chain rule is used for composition functions:
fog
( )
x
( ) =
f g x
( )
( )
B.
Method 1: Rewrite the function in terms of
y=f(u)
and
u=g(x).
If
y= f(u)
and
u=g(x),
then
dy
dx
=
dy
du
!
"
#
$
du
dx
!
"
#
$
.
C.
Method 2: If
y
=
fog
( )
x
( ) =
f g x
( )
( )
, then
!
y
=
!
f
g x
( )
( )
!
g
x
( )
D.
Example: The radius of a right circular cylinder is increasing at a rate of
2
in
min
and the
height is increasing at
3
in
min
. How fast is the lateral surface area changing when the radius
is 10 inches and the height is 12 inches?
II.
The Chain Rule for Functions of Two or More Variables
A. Functions of Two Variables
1. Theorem 5: The Chain Rule for Functions of Two Independent Variables
If
z
=
f
(
x
,
y
)
has continuous partial derivatives
f
x
and
f
y
and if
x
=
x
(
t
),
y
=
y
(
t
)
are
differentiable functions of
t
, then the composite
z
=
f
(
x
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This note was uploaded on 04/05/2008 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.
 Spring '03
 MECothren
 Chain Rule, Multivariable Calculus, The Chain Rule

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